Difference between revisions of "2005 AMC 10B Problems/Problem 24"

Revision as of 16:04, 16 December 2021

Problem

Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$ . The integers $x$ and $y$ satisfy $x^2 - y^2 = m^2$ for some positive integer $m$ . What is $x + y + m$ ?

$\textbf{(A) } 88 \qquad \textbf{(B) } 112 \qquad \textbf{(C) } 116 \qquad \textbf{(D) } 144 \qquad \textbf{(E) } 154$

Solution 1

Let $x = 10a+b, y = 10b+a$ . The given conditions imply $x>y$ , which implies $a>b$ , and they also imply that both $a$ and $b$ are nonzero.

Then, $x^2 - y^2 = (x-y)(x+y) = (9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2$ .

Since this must be a perfect square, all the exponents in its prime factorization must be even. $99$ factorizes into $3^2 \cdot 11$ , so $11|(a-b)(a+b)$ . However, the maximum value of $a-b$ is $9-1=8$ , so $11|a+b$ . The maximum value of $a+b$ is $9+8=17$ , so $a+b=11$ .

Then, we have $33^2(a-b) = m^2$ , so $a-b$ is a perfect square, but the only perfect squares that are within our bound on $a-b$ are $1$ and $4$ . We know $a+b=11$ , and, for $a-b=1$ , adding equations to eliminate $b$ gives us $2a=12 \Longrightarrow a=6, b=5$ . Testing $a-b=4$ gives us $2a=15 \Longrightarrow a=\frac{15}{2}, b=\frac{7}{2}$ , which is impossible, as $a$ and $b$ must be digits. Therefore, $(a,b) = (6,5)$ , and $x+y+m=\boxed{\textbf{(E) }154}$ .

Solution 2

The first steps are the same as Solution 1. Let $x = 10a+b, y = 10b+a$ , where we know that a and b are digits (whole numbers less than $10$ ).

Like Solution 1, we end up getting $(9a - 9b)(11a + 11b) = 99(a-b)(a+b) = m^2$ . This is where the solution diverges.

We know that the left side of the equation is a perfect square because $m$ is an integer. If we factor $99$ into its prime factors, we get $3^2\cdot 11$ . In order to get a perfect square on the left side, $(a-b)(a+b)$ must make both prime exponents even. Because the a and b are digits, a simple guess would be that $(a+b)$ (the bigger number) equals $11$ while $(a-b)$ is a factor of nine (1 or 9). The correct guesses are $a = 6, b = 5$ causing $x = 65, y = 56,$ and $m = 33$ . The sum of the numbers is $\boxed{\textbf{(E) }154}$

Solution 3

Once again, the solution is quite similar as the above solutions. Since $x$ and $y$ are two digit integers, we can write $x = 10a+b, y = 10b+a$ and because $x^2 - y^2 = (x-y)(x+y)$ , substituting and factoring, we get $99(a+b)(a-b) = m^2$ .

Therefore, $(a+b)(a-b) = \frac{m^2}{99}$ and $\frac{m^2}{99}$ must be an integer. A quick strategy is to find the smallest such integer $m$ such that $\frac{m^2}{99}$ is an integer. We notice that 99 has a prime factorization of $3^2 \cdot 11.$

Let $m^2 = n.$ Since we need a perfect square and 3 is already squared, we just need to square 11. So $3^2 \cdot 11^2$ gives us 1089 as $n$ and $m = \sqrt{1089} = 33.$ We now get the equation $(x-y)(x+y) = 33^2$ , which we can also write as $(x-y)(x+y) = 11^2 \cdot 3^2$ .

A very simple guess assumes that $x-y=3^2$ and $x+y=11^2$ since $x$ and $y$ are positive. Finally, we come to the conclusion that $x=65$ and $y=56$ , so $x+y+m$ $=$ $\boxed{\textbf{(E) }154}$ .

Note that all of the solutions used $a+b$ or $a-b$ as part of their solution.

Solution 4

Continue the same as Solution $3$ until we get $33$ . Knowing that $33^2 = 1089$ , we have narrowed down our Pythagorean triples. We know that the $2$ other squares should be larger than $33^2$ , so we can start testing.

If we start testing the $40$ s, it is fruitless since the closest to $33^2$ would be $33 - 45 - 54$ which is not a Pythagorean triple. We can start by testing out the $50$ s, and it turns our that $33 - 56 - 65$ is a Pythagorean triple. Therefore, our answer is $33+56+65$ = $\boxed{\textbf{(C) }144}$ .

~Arcticturn

@@ Line 34: / Line 34: @@
 == Solution 4 ==
-Continue the same as solution <math>3</math> until we get <math>33</math>. Knowing that <math>33^2 = 1089</math>, we have narrowed down our Pythagorean triples. We know that the <math>2</math> other squares should be larger than <math>33^2</math>, so we can start testing. If we start testing the <math>40</math>s, it is fruitless since the closest to <math>33^2</math> would be <math>33 - 45 - 54</math> which is not a Pythagorean triple. We can start by testing out the <math>50</math>s, and it turns our that <math>33 - 56 - 65</math> is a Pythagorean triple. Therefore, our answer is <math>33+56+65</math> = $\boxed {(C)144}.
+Continue the same as Solution <math>3</math> until we get <math>33</math>. Knowing that <math>33^2 = 1089</math>, we have narrowed down our Pythagorean triples. We know that the <math>2</math> other squares should be larger than <math>33^2</math>, so we can start testing.
+If we start testing the <math>40</math>s, it is fruitless since the closest to <math>33^2</math> would be <math>33 - 45 - 54</math> which is not a Pythagorean triple. We can start by testing out the <math>50</math>s, and it turns our that <math>33 - 56 - 65</math> is a Pythagorean triple. Therefore, our answer is <math>33+56+65</math> = <math>\boxed{\textbf{(C) }144}</math>.
 ~Arcticturn

2005 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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